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A calculus exam focusing on topics such as differentials, gradients, tangent planes, and extreme values. It includes various problems involving differentiating equations, finding gradients and relative directions, determining tangent planes and normal lines, and identifying critical points and extreme values. Students are required to show their work and simplify where necessary.
Typology: Exams
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Name: Box #
Instructions
ï Answer all the questions directly on the test.
ï This is a paper and pencil only exam
ï Show all the necessary work, and where required, write your answers out neatly in English sentences.
ï Make reasonable simplifications.
Question Possible Points Points Obtained 1 25 2 25 3 25 4 25 Total 100
1 R
1.a By differentiating both side of the equation above show that
1 R^2
1.b By using equation 1.1 and its for R 2 above find a formula for dR in terms of R 1 , R 2 , dR 1 , dR 2
dR =
1.c Suppose that R 1 and R 2 1. 0000 ◊ 105 and 2. 0000 ◊ 105 so that the computed value of R is ohms 0. 6667 ◊ 105 ≈ 23 ◊ 105. Suppose that the maximum error of both R 1 and R 2 is 100 ohms. Using differentials, estimate the maximum error of the computed value of R.
3.a Consider the quadratic surface Q 1 given by:
x^2 − xy + y^2 − yz + z^2 + x + y + z = 6
What is the equation of the tangent plane to the surface at P 0 = (1, − 1 , 1)?
3.b Consider the sphere centered at the origin radius
3 , which also passes through the point P 0. What is the angle between the normal lines to the two surfaces at P 0?
3.c Are the two surfaces orthogonal or tangent? Justify your answer.
4.a Find all the critical points the given domain.
